Ahmes (more accurately Ahmose) was an ancient Egyptian scribe who lived during the Second Intermediate Period and the beginning of the Eighteenth Dynasty (the first dynasty of the New Kingdom).^[1] He is best known for his work in mathematics.

[edit] Background

The mortuary temple's entrance from within the Pyramid of Djoser complex at Saqqara, where Ahmes left graffiti at the North Chapel to commemorate his visit

Ahmes was the son of the scribe Iyptah.^[1]

Alongside a scribal companion from the school of Sethemhab, Ahmes once traveled to Saqqara, Egypt, to visit the Old Kingdom monument of the Temple of Djoser. His companion describes Ahmes's delight with the ancient site in graffiti that they personally inscribed at the North Chapel of the temple complex:

The scribe Ahmose came to see the Temple of Djoser. He found it as if heaven were within it, Ra rising in it. Then he said: 'Let loaves and oxen and fowl and good and pure things fall to the Ka of the justified Djoser: may heaven rain fresh myrrh, may it drip incense!'^[2]

[edit] Works

A surviving work of Ahmes is part of the Rhind Mathematical Papyrus now located in the British Museum (Newman, 1956). Ahmes states that he copied the papyrus from a now-lost Middle Kingdom original, dating around 1650 BC. The work is entitled Directions for Knowing All Dark Things and is a collection of problems in arithmetic, algebra, geometry, weights and measures, business and recreational diversions. The 51 member 2/nth table and the following 87 problems were presented with red auxiliary numbers. Ahmes offered brief notes for hard-to-read red auxiliary number steps for 20th century scholars. By using additional documents like the Akhmim Wooden Tablet, Egyptian Mathematical Leather Roll, Reisner Papyrus and the Moscow Mathematical Papyrus, a theoretical view of Ahmes' arithmetic was found after 2002, and formally published in 2006, with updated arithmetic operations reported. For example, the 2/nth table generally converted 1/p, 1/pq, 2/p, 2/pq by selecting optimized red auxiliary (Least Common Multiple or LCM) numbers, scaling the vulgar fractions, facilitating an aliquot part method, to write out exact unit fraction series. The generalized LCM method was discussed in the 1202 AD Liber Abaci in a subtraction context. Red auxiliary LCM conversion method was employed by Ahmes in the RMP 2/n table, the EMLR, and all other Middle Kingdom mathematical texts.

A portion of the Rhind Mathematical Papyrus

On a broader level, considering the RMP and its parent documents, like the Akhmim Wooden Tablet and the Kahun Papyrus the Egyptian fraction notation was developed around 2000 BCE. The Egyptian fraction notation replaced the Old Kingdom's binary fraction Horus-Eye notation, an awkward round-off system, and the Old Kingdom form of multiplication (used by Ahmes as a proof method). Middle Kingdom scribes developed Egyptian multiplication and division to replace the Old Kingdom multiplication operation, and its unknown form of division. Finally, considering weights and measures connections provided by the Middle Kingdom texts, the why's and how's of proto-number theory that Ahmes drew upon has come into focus. Ahmes' methods, as taught to him, and followed by later scribes, wrote vulgar fractions in exact ways, never rounding off when rational numbers were involved. When the Egyptian fraction methods are stripped away, easy-to-read rational number arithmetic is exposed, revealing modern looking arithmetic operations.

Yet there were times when Ahmes did make use of approximations. For example, Ahmes states without proof that a circular field with a diameter of 9 units is equal in area to a square with sides of 8 units (Beckmann, 1971). In modern notation, Ahmes' method implies the following approximation:

This method leads to a value of pi approximately equal to 3.16049, which comes within 0.6% of the true value of pi. This approximation for pi as an irrational number reached well beyond the rational number domain of Egyptian mathematics. It was used to compute the volume of a hekat, and its many sub-units, including the hin, ro, and dja as recorded in the RMP, and the Akhmim Wooden Tablet. This is known to be the first historical reference to pi.

[edit] Notes

[edit] References

• Allen, Don. April 2001. The Ahmes Papyrus and Summary of Egyptian Mathematics.
• Borbola J. Kiralykörök /the Hungarian reading and solving of the Rhind-papyrus/
• Borbola J. Olvassunk együtt magyarul /Hungarian reading and solving of the Moskow Mathematic Papyrus/
• Chace, Arnold Buffum. 1927-1929. The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations. Classics in Mathematics Education 8. 2 vols. Oberlin: Mathematical
• Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society. ISBN 0-87169-232-5
• Gardner, Milo, "An Ancient Egyptian Problem and its Innovative Arithmetic Solution", Ganita Bharati, 2006, Vol 28, Bulletin of the Indian Society for the History of Mathematics, MD Publications, New Delhi, pp 157-173
• Ahmes Bird Feeding Rate Method PlanetMath, 2008.
• Gillings, R., Mathematics in the Time of the Pharaohs. Boston, MA: MIT Press, pp. 202-205, 1972. ISBN 0-262-07045-6. (Out of print) Association of America. (Reprinted Reston: National Council of Teachers of Mathematics, 1979). ISBN 0-87353-133-7
• Joseph, George G., Crest of the Peacock, Penguin Books, 1991 (pages 71-73)
• Imhausen, A., Ägyptische Algorithmen. Eine Untersuchung zu den mittelägyptischen mathematischen Aufgabentexten, Wiesbaden 2003.
• Peet, Thomas Eric. 1923. The Rhind Mathematical Papyrus, British Museum 10057 and 10058. London: The University Press of Liverpool limited and Hodder & Stoughton limited
• "Arithmetic in the Middle Kingdom." J. Egyptian Arch. 9, 91-95, 1923
• O'Connor and Robertson, 2000. Mathematics in Egyptian Papyri.* Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British Museum Publications Limited. ISBN 0-7141-0944-4
• Truman State University, Math and Computer Science Division. Mathematics and the Liberal Arts: The Rhind/Ahmes Papyrus.
• Williams, Scott W. Mathematicians of the African Diaspora, containing a page on Egyptian Mathematics Papyri.
• Wolfram, "Rhind Papyrus". MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/RhindPapyrus.html.

[edit] External links

• Breaking the RMP 2/n Table Code
• http://mathforum.org/kb/message.jspa?messageID=6579539&tstart=0 Math forum and two ways to calculate 2/7
• http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html New and Old Ahmes Papyrus classifications